基于门限算法的SM2协同签名算法分析-openssl源码验证

介绍

门限密码体制 是安全多方计算中对抗合谋攻击的一个重要工具，在门限密码体制中， n个参与者联合生成公钥，解密密钥由 n 个参与者联合持有公可以直接加密消息但解密需要 n个参与者中一定数量人员参与才能正确解密.
接下来介绍一种基于门限密码思想的SM2协同签名方法，是一种（2，2）门限密码方案，即需要两方参与的协同签名流程。
参考文献《基于SM2的双方共同签名协议及其应用》
该方法主要分两个流程，协同公钥生成和协同签名。

1 协同公钥生成流程

双方主要交互流程如下图：

2 协同签名流程

双方主要交互流程如下图：

3 正确性分析

4 使用openssl接口进行代码测试

测试数据如下：

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生成协同公钥：
=======================================
发送方A：
Da 密钥分量:
0xd4, 0xf3, 0x1d, 0x15, 0x17, 0x07, 0x51, 0x33, 0x72, 0xd7, 0x5e, 0x56, 0x26, 0x61, 0xf7, 0x9d, 
0x95, 0x61, 0xe8, 0x8a, 0xa3, 0xb9, 0xa9, 0xf7, 0x67, 0xaf, 0x23, 0x95, 0x5e, 0xbb, 0xe2, 0x82, 
公钥分量Pka:
-X
0x06, 0x00, 0x38, 0x73, 0x0e, 0xfe, 0x5b, 0x48, 0x28, 0x6b, 0x97, 0xd7, 0xff, 0x1e, 0xbb, 0xe7, 
0x55, 0x0c, 0xe4, 0x5d, 0x94, 0x34, 0x4f, 0x15, 0x50, 0xa6, 0x63, 0x10, 0x37, 0xd7, 0x3c, 0xe8, 
-Y
0xbc, 0x42, 0x36, 0xcf, 0xf3, 0x41, 0xb1, 0x23, 0xf7, 0x49, 0x38, 0x1a, 0x4c, 0x64, 0xbc, 0x81, 
0x84, 0x3b, 0xf0, 0xc4, 0x5c, 0x38, 0xde, 0x2d, 0x82, 0xfe, 0x46, 0xa6, 0xbf, 0x1c, 0xe8, 0xfc, 
发送方B：
Db 密钥分量:
0xca, 0xe1, 0x6c, 0x68, 0x4b, 0x21, 0xad, 0x08, 0x4c, 0xd3, 0x01, 0x5c, 0xdb, 0xfa, 0xa8, 0x62, 
0x6e, 0x89, 0xd3, 0x1f, 0xa8, 0x01, 0x04, 0xe4, 0x13, 0x6c, 0x75, 0xa9, 0xca, 0x4d, 0x44, 0xa9, 
公钥分量Pkb:
-X
0xc7, 0x11, 0x15, 0x28, 0xc8, 0x31, 0x1f, 0xe9, 0x96, 0xb6, 0x6a, 0x2f, 0x07, 0xbe, 0x69, 0x30, 
0x4a, 0x9f, 0x22, 0x4c, 0x23, 0x40, 0xe9, 0x2a, 0x0e, 0x6c, 0x15, 0x21, 0x62, 0x2e, 0xcb, 0xa6, 
-Y
0x7d, 0xfd, 0xa6, 0x61, 0x45, 0xe4, 0x1f, 0x22, 0xa5, 0x28, 0xfb, 0x8e, 0x04, 0x2b, 0x0b, 0x13, 
0xc3, 0x30, 0xda, 0x74, 0x05, 0x8e, 0x9c, 0x2e, 0xe1, 0xbf, 0x07, 0xc8, 0x77, 0x2d, 0x08, 0xf9, 
共同公钥Pk：
0x04, 0x1b, 0xde, 0x21, 0x21, 0x98, 0xb4, 0xb9, 0x6d, 0x24, 0x72, 0x5e, 0x57, 0x51, 0x3c, 0x95, 
0xf9, 0xb7, 0x3c, 0x91, 0x2a, 0x7a, 0x6a, 0x1f, 0x82, 0xf6, 0x80, 0x8a, 0x2d, 0x79, 0x51, 0xa1, 
0xc6, 0xca, 0xd8, 0x89, 0xf2, 0x91, 0x3b, 0x0f, 0xd7, 0xe1, 0x1c, 0x55, 0xe7, 0xa9, 0x98, 0xcc, 
0x65, 0x5e, 0x1c, 0x46, 0x17, 0xd7, 0x33, 0x8d, 0x90, 0x62, 0x80, 0x80, 0xff, 0xb9, 0xaa, 0x03, 
0xe5, 
=======================================
协同签名：
=======================================
发送方A：
Ka:
0x4c, 0xa0, 0xf7, 0xae, 0xe8, 0x18, 0x79, 0x59, 0x88, 0x4d, 0x2b, 0x1a, 0x78, 0xca, 0xa9, 0x4c, 
0xb1, 0x70, 0x5f, 0x0b, 0x11, 0x31, 0x0f, 0xa3, 0xb4, 0x1c, 0xfe, 0x2b, 0x15, 0xc7, 0x05, 0xa8, 
Ra:
-X
0x87, 0xca, 0xb9, 0x85, 0xe4, 0xda, 0xcc, 0xd6, 0xe1, 0x8d, 0xba, 0xed, 0x7e, 0x6d, 0x8e, 0xd5, 
0x80, 0xd4, 0x29, 0xac, 0x1e, 0x83, 0xb3, 0x13, 0x94, 0xb8, 0x82, 0x0f, 0x0b, 0x52, 0xf9, 0x47, 
-Y
0xf8, 0xb0, 0xd8, 0x75, 0xd5, 0xf7, 0x79, 0x9e, 0xce, 0xa1, 0x44, 0x7f, 0x79, 0xdb, 0x1a, 0x31, 
0x37, 0x1d, 0x0f, 0xaf, 0x25, 0xad, 0xb6, 0x42, 0x25, 0x7a, 0xb5, 0xcd, 0xda, 0x81, 0x68, 0x42, 
Ra`:
-X
0xb8, 0xcb, 0xe1, 0x48, 0x05, 0x7d, 0xaf, 0x00, 0xb6, 0x58, 0x36, 0xcd, 0x47, 0xd0, 0x09, 0x37, 
0xe3, 0x00, 0x40, 0x9e, 0x0a, 0xd1, 0x5a, 0x62, 0x50, 0xac, 0x06, 0x2e, 0x1e, 0x75, 0x50, 0xf5, 
-Y
0x78, 0xf6, 0x85, 0xca, 0x70, 0x56, 0xaa, 0x5a, 0x67, 0x2c, 0x6a, 0xf2, 0x73, 0xf6, 0xbd, 0xc4, 
0xa7, 0x35, 0x17, 0x84, 0x4d, 0xd4, 0x82, 0x37, 0xa9, 0xfb, 0x72, 0x08, 0x30, 0xab, 0x3e, 0x3d, 
发送方B：
point Ra verify sucessful
Kb:
0x2c, 0xe9, 0xf7, 0x07, 0x21, 0x87, 0xd6, 0x56, 0x50, 0xf9, 0x4a, 0x5c, 0x43, 0x26, 0x2e, 0x33, 
0x83, 0xfc, 0x0d, 0xf5, 0xa4, 0x0c, 0x3d, 0x49, 0xe1, 0xff, 0x18, 0xd4, 0x5c, 0xe6, 0x1d, 0xee, 
Rb:
-X
0x98, 0xc7, 0xdf, 0xaa, 0x41, 0xaf, 0x2a, 0xb9, 0x34, 0x30, 0xc9, 0x7d, 0xda, 0xad, 0x8c, 0xd2, 
0x3a, 0xa1, 0x71, 0x3e, 0xda, 0x33, 0x1f, 0xa5, 0x16, 0xd7, 0x6c, 0xf7, 0x56, 0xe4, 0xee, 0xa6, 
-Y
0x62, 0xce, 0x50, 0xc7, 0x1e, 0x5e, 0xeb, 0x90, 0x29, 0x93, 0x0a, 0xfb, 0x69, 0x07, 0x29, 0x19, 
0xa8, 0xd7, 0x6d, 0xdc, 0x5c, 0x10, 0x23, 0x69, 0xa4, 0x7f, 0xbe, 0x78, 0xf2, 0x35, 0xfe, 0x46, 
Rb`:
-X
0x5f, 0xeb, 0x24, 0x52, 0xdf, 0x08, 0xd2, 0xda, 0xac, 0x48, 0x4d, 0xfe, 0x3f, 0x44, 0x03, 0x55, 
0xaf, 0x67, 0xde, 0x60, 0x98, 0x80, 0xbb, 0xa8, 0xfd, 0x9c, 0xaa, 0x6b, 0x86, 0x58, 0xa0, 0x54, 
-Y
0x56, 0x6a, 0x6b, 0xdc, 0x76, 0xcb, 0x19, 0x0c, 0x31, 0xce, 0xad, 0xb8, 0x5b, 0x28, 0xe0, 0x38, 
0x3c, 0x32, 0x59, 0x0c, 0x88, 0x54, 0xcf, 0x9b, 0xfd, 0xe6, 0x2e, 0x6a, 0xfd, 0xf1, 0x01, 0x56, 
发送方A：
消息M:
0x01, 0x02, 0x03, 0x04, 0x05, 0x06, 0x07, 0x08, 0x09, 
Z值:
0x31, 0x32, 0x33, 0x34, 0x35, 0x36, 0x37, 0x38, 0x31, 0x32, 0x33, 0x34, 0x35, 0x36, 0x37, 0x38, 
e:
0xfe, 0x7c, 0x9f, 0x48, 0x39, 0x4a, 0xa7, 0xfa, 0x0a, 0x0f, 0x99, 0x9e, 0x1e, 0xe1, 0x41, 0xb5, 
0x5a, 0xbe, 0x20, 0xc5, 0x59, 0x47, 0x13, 0x6b, 0xaa, 0x9e, 0x52, 0x1b, 0x31, 0x77, 0x76, 0x7b, 
point Rb verify sucessful
点R (xa,ya):
-X
0xa0, 0x73, 0x36, 0x10, 0xd1, 0x74, 0x5d, 0xcd, 0x77, 0xb4, 0x5b, 0x5e, 0xa2, 0x0e, 0xba, 0xdc, 
0xc5, 0xd9, 0x9c, 0x34, 0xe2, 0x9e, 0xb9, 0xe9, 0x59, 0xb0, 0xa4, 0x90, 0xd6, 0x23, 0x13, 0x89, 
-Y
0xde, 0x3c, 0x28, 0xa9, 0xfc, 0xbd, 0x1f, 0x32, 0xc6, 0x1a, 0xd7, 0x63, 0x16, 0xa7, 0x0f, 0x20, 
0x41, 0x67, 0x29, 0x3f, 0x53, 0xb2, 0x2b, 0x07, 0x5d, 0x2d, 0x5e, 0x9d, 0xce, 0x62, 0xdf, 0x0e, 
r:
0x9e, 0xef, 0xd5, 0x5a, 0x0a, 0xbf, 0x05, 0xc7, 0x81, 0xc3, 0xf4, 0xfc, 0xc0, 0xef, 0xfc, 0x92, 
0xae, 0x93, 0xdd, 0x8f, 0x1a, 0x1f, 0xc8, 0x29, 0xb0, 0x93, 0x02, 0xa2, 0xcd, 0xc5, 0x48, 0xe1, 
s`:
0x91, 0xa1, 0xd9, 0x3f, 0x0f, 0x9b, 0xd0, 0xf6, 0x87, 0x4d, 0xdd, 0x98, 0x1e, 0xf1, 0x98, 0xa3, 
0xc8, 0xbc, 0xce, 0x21, 0xd3, 0x99, 0xdd, 0x02, 0x33, 0x9c, 0xc5, 0xdf, 0xaa, 0xfd, 0x3d, 0xcd, 
发送方B：
t:
0xc7, 0xd6, 0x11, 0xd0, 0x79, 0xed, 0xb7, 0x91, 0x0c, 0x06, 0xb5, 0xe8, 0xf7, 0xe6, 0x44, 0xc8, 
0xe2, 0x48, 0x97, 0xa4, 0xa6, 0x44, 0xbe, 0x04, 0x2a, 0x90, 0x5e, 0x36, 0x3c, 0x7f, 0x0a, 0x4b, 
发送方A：
r:
0x9e, 0xef, 0xd5, 0x5a, 0x0a, 0xbf, 0x05, 0xc7, 0x81, 0xc3, 0xf4, 0xfc, 0xc0, 0xef, 0xfc, 0x92, 
0xae, 0x93, 0xdd, 0x8f, 0x1a, 0x1f, 0xc8, 0x29, 0xb0, 0x93, 0x02, 0xa2, 0xcd, 0xc5, 0x48, 0xe1, 
s:
0x28, 0xe6, 0x3c, 0x76, 0x6f, 0x2e, 0xb1, 0xc9, 0x8a, 0x42, 0xc0, 0xec, 0x36, 0xf6, 0x48, 0x36, 
0x33, 0xb4, 0xba, 0x15, 0x8c, 0x24, 0xf5, 0xda, 0x79, 0xfd, 0x5b, 0x93, 0x6e, 0xb9, 0xc1, 0x6a, 
r:
0x9e, 0xef, 0xd5, 0x5a, 0x0a, 0xbf, 0x05, 0xc7, 0x81, 0xc3, 0xf4, 0xfc, 0xc0, 0xef, 0xfc, 0x92, 
0xae, 0x93, 0xdd, 0x8f, 0x1a, 0x1f, 0xc8, 0x29, 0xb0, 0x93, 0x02, 0xa2, 0xcd, 0xc5, 0x48, 0xe1, 
s:
0x28, 0xe6, 0x3c, 0x76, 0x6f, 0x2e, 0xb1, 0xc9, 0x8a, 0x42, 0xc0, 0xec, 0x36, 0xf6, 0x48, 0x36, 
0x33, 0xb4, 0xba, 0x15, 0x8c, 0x24, 0xf5, 0xda, 0x79, 0xfd, 0x5b, 0x93, 0x6e, 0xb9, 0xc1, 0x6a, 
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经过和标准SM2签名对比，可以验签通过：

验签 通过
算法：  SM2公钥验签
数据  长度9(0x09)字节
01 02 03 04 05 06 07 08 09 
签名结果  长度64(0x40)字节
9E EF D5 5A 0A BF 05 C7 81 C3 F4 FC C0 EF FC 92 AE 93 DD 8F 1A 1F C8 29 B0 93 02 A2 CD C5 48 E1 28 E6 3C 76 6F 2E B1 C9 8A 42 C0 EC 36 F6 48 36 33 B4 BA 15 8C 24 F5 DA 79 FD 5B 93 6E B9 C1 6A 
ID：16(0x10)字节
31 32 33 34 35 36 37 38 31 32 33 34 35 36 37 38 
HASH：32(0x20)字节
FE 7C 9F 48 39 4A A7 FA 0A 0F 99 9E 1E E1 41 B5 5A BE 20 C5 59 47 13 6B AA 9E 52 1B 31 77 76 7B 
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5 测试源码

https://download.csdn.net/download/qq_39952971/87966483